Problem: What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5),$ and the other on the sphere of radius 87 with center $(12,8,-16)$?
Explanation: Let $O$ be center of the first sphere, and let $P$ be the center of the second sphere.  Then
\[OP = \sqrt{(-2 - 12)^2 + (-10 - 8)^2 + (5 - (-16))^2} = 31.\][asy]
unitsize(1 cm);

pair A, B, O, P;

O = (0,0);
P = 8*dir(15);
A = dir(195);
B = P + 2*dir(15);

draw(Circle(O,1));
draw(Circle(P,2));
draw(A--B);

label("$A$", A, W);
label("$B$", B, E);
dot("$O$", O, S);
dot("$P$", P, S);
[/asy]

Let $A$ be a point on the first sphere, and let $B$ be a point on the second sphere.  Then by the Triangle Inequality,
\[AB \le AO + OP + PB = 19 + 31 + 87 = 137.\]We can achieve this by taking $A$ and $B$ to be the intersections of line $OP$ with the spheres, as shown above.  Hence, the largest possible distance is $\boxed{137}.$